Geodesic image regression with a sparse parameterization of diffeomorphisms James Fishbaugh 1 Marcel Prastawa 1 Guido Gerig 1 Stanley Durrleman 2 1 Scientific Computing and Imaging Institute, University of Utah 2 INRIA/ICM, Pitié Salpêtrière Hospital, Paris, France

Image Regression 6 months12 months18 months8 months14 months16 months10 months Why image regression? Extrapolation for change prediction Align images and cognitive scores acquired at different times Align subjects with scans acquired at different times Improved understanding of normal and pathological brain changes 1 of 18

Previous Work Kernel regressionGeodesic regression Davis et al. ICCV 2007Niethammer et al. MICCAI 2011Singh et al. ISBI 2013 Require to store many model parameters ~ number of voxels Image evolution described by considerably fewer parameters Concentrated in areas undergoing most dynamic changes 2 of 18

Motivation for Sparsity Fewer parameters Location of parameters Potential for greater statistical power – less noise in description Concentrated in areas undergoing the most dynamic changes Number of parameters should reflect complexity of anatomical changes, not the sampling of the images Localize potential biomarkers 3 of 18

Compact and generative statistical model of growth Geodesic Image Regression Geodesic path on a sub-group of diffeomorphisms (Dupuis 98, Trouvè 95,98) 4 of 18

Geodesic Image Regression S 0 = {c 0, α 0 } I0I0 O1O1 O3O3 O2O2 5 of 18

Geodesic shooting to evolve control points S 0 = {c 0, α 0 } Methods: Shooting 6 of 18

Trajectory of control points defines flow of diffeomorphisms Physical pixel coordinates y follow the trajectory which evolves in time as Methods: Flow (5, 5, 60.25) Deformed images constructed by interpolation 7 of 18

Summary Of Method 1) Shoot control points2) Trajectory defines flow 3) Flow pixel locations4) Interpolate in baseline image 8 of 18

Subject to Shoot Flow Regression Criterion 9 of 18

Method Overview Gradient with respect to control points and initial momenta Gradient with respect to initial image Gradient of Regression Criterion 1) Flow voxel Y k (t) to time t and compute residual 2) Grey value in residual is distributed to neighboring voxels with weights from trilinear interoplation 3) Grey values accumulated for every observed image 10 of 18

Method Overview Sparsity on Initial Momenta Fast Iterative Shrinkage-Thresholding Algorithm (Beck 09)  Use previous gradient of criterion without L 1 penalty  Threshold momentum vectors with small magnitude Select a small subset of momenta which best describe the dynamics of image evolution 11 of 18  Used in context of atlas building (Durrleman 12,13)

Synthetic Evolution (2D) Generated by shooting baseline with 79,804 predefined momenta Time 1Time 2Time 3Time 4Time 5 Impact of sparsity parameter on model estimation 12 of 18

Synthetic Evolution (2D) From 79,804 to 67 momenta 13 of 18

Pediatric Brain Development (2D) Models estimated backwards in time with varying sparsity T1W image of same child over time 14 of 18

Method Overview Pediatric Brain Development (2D) From 45,435 to 47 momenta 15 of 18

Brain Atrophy in Alzheimer's Disease (3D) T1W image of same patient over time years71.38 years71.78 years72.79 years Six years predicted brain atrophy with 35,937 momenta 98% decrease in number of parameters 16 of 18

Conclusions Geodesic image regression framework:  Decouples deformation parameters from image representation  L 1 penalty which selects optimal subset of initial momenta Number of parameters reduced with only minimal cost in terms of matching target data Future work:  Kernels at multiple scales (Sommer 11)  Other image matching metrics, LCC (Avants 07, Lorenzi 13)  Combine with a framework for longitudinal analysis 17 of 18

This work was supported by: NIH (NINDS) 1 U01 NS (4D shape HD) NIH (NICHD) RO1 HD (ACE, project IBIS) NIH (NIBIB) 2U54 EB (NA-MIC) Acknowledgments Thank you 18 of 18